Proportion is a concept of Mathematics which gives the relation between any two mathematical quantities. Two quantities are said to be proportional if they are multiplicatively connected by a constant. The proportional relationship between any two quantities can also be defined as the quantities whose product or ratio is constant. Two quantities are said to be directly proportional if their ratio is constant. If the product of any two quantities is a constant, then those two quantities are said to be inversely proportional. The two quantities which are directly proportional are related by a Direct Proportion symbol ‘∝’. The symbol for proportionality is removed by adding a Direct Proportion constant.

Direct Proportion meaning is explained as follows. Two measurable quantities are said to be directly proportional if the increase in one quantity results in the increase of the other quantity and vice versa. In direct variation, the ratio of two measurable quantities is constant. For example, if x and y are the two measurable quantities that are directly proportional to each other, then the direct proportion definition is Mathematically written as x ∝ y. If the direct proportion symbol is to be removed, a proportionality constant is added and the direct proportion symbol is replaced by an equal sign.

x = k y

\[\frac{{x }}{y}\] = K

In the above equation, ‘k’ is a proportionality constant.

If x1 and y1 are the initial values of any two quantities that are directly proportional to each other and x2 and y2 are the final values of those quantities. Then according to the direct proportionality relationship,

\[\frac{{x1}}{y1}\] = k and \[\frac{{x2 }}{y2}\] = k

So, we can infer that the ratio of initial values and the final values of any two quantities varying directly are equal and constant.

\[\frac{{x1}}{y1}\] = \[\frac{{x2}}{y2}\] = k

Marks scored is directly proportional to the performance in the test.

Temperature is directly proportional to heat.

Energy is directly proportional to work.

Speed is directly proportional to distance.

Earning is directly proportional to the amount of work done.

The amount of food we consume is directly proportional to how hungry we are.

These are just a few real-world Direct Proportion examples.

In one of the real situations of direct proportion examples, a bus travels 150 km in 5 hours. What is the time taken by the bus to travel 700 km?

Solution:

Distance traveled and time taken are directly proportional to each other.

In the given question, the distance traveled in case 1 is x1 = 150 km

The distance traveled in case 2 is x2 = 700 km

The time taken in case 1 is y1 = 5 hours

Time taken in case 2 is y2 = ?

The proportionality relationship can be stated as:

\[\frac{{x1}}{y1}\] = \[\frac{{x2}}{y2}\]

\[\frac{{150}}{5}\] = \[\frac{{700}}{y2}\]

y2 = \[\frac{{700}}{150}\] ✕ 5

y2=23.33

So, the time taken by the bus to travel 700 km is 23.33 hrs

Given that a and b are directly proportional to each other, complete the table given below.

Solution:

From the table x1 = 4, y1 = 6, x2 = 5, x3 = 12, x4 = 6

y2 = ? y3 = ? y4 = ?

Case 1: To find y2

\[\frac{{x1}}{y1}\] = \[\frac{{x2}}{y2}\]

\[\frac{{4}}{6}\] = \[\frac{{5}}{y2}\]

y2= \[\frac{{5}}{4}\] ✕ 6

y2=7.5

Case 2: To find y3

\[\frac{{x1}}{y1}\] = \[\frac{{x3}}{y3}\]

\[\frac{{4}}{6}\] = \[\frac{{12}}{y3}\]

y3 = \[\frac{{12}}{4}\] ✕ 6

y3= 18

Case 3: To find y4

\[\frac{{x1}}{y1}\] = \[\frac{{x4}}{y4}\]

\[\frac{{4}}{6}\] = \[\frac{{6}}{y4}\]

y4 = \[\frac{{6}}{4}\] ✕ 6

y4 =9

So, the completed table is as below:

Sumanth has Rs. 400/- with him. If he can purchase 5 kgs of ghee for 2180, how much ghee can he purchase with the amount he has?

Solution:

Total amount for 5 kg ghee is x1 = Rs. 2180/-

Ghee purchased with Rs. 2180/- is y1 = 5 kg

Amount with Sumanth is x2 = Rs. 400/-

Ghee purchased with Rs. 400/- is y2 = ?

The money and amount of ghee purchased are directly proportional to each other.

The direct proportionality relationship can be written as:

\[\frac{{x1}}{y1}\] = \[\frac{{x2}}{y2}\]

\[\frac{{2180}}{5}\] = \[\frac{{400}}{y2}\]

y2= \[\frac{{400}}{2180}\] ✕ 5

y2= \[\frac{{400}}{2180}\] ✕ 5

y2=0.917

Sumanth can purchase 0.917 kgs of ghee with Rs. 400.

Time and work are directly proportional to each other. Is this statement true?

Yes

No

Which of the following are directly proportional measurements?

Current flow and resistance

Volume and temperature

Mass and weight

From the given figure, identify the graphs that indicate direct proportion definition.

FAQ (Frequently Asked Questions)

What is Mathematical proportion?

Any two physical quantities that vary is said to be proportional according to Mathematics if they are multiplicatively connected to each other by a constant term. For example, the more we eat, the more energy we gain, and the more we run, the more energy we lose.

There are two kinds of proportionalities in Mathematics. They are:

Direct proportion meaning: Two quantities are said to be directly proportional to each other if the ratio of their values is constant at any instant of time. The increase in one quantity results in the increase of the other, For example, the more we exercise, the more fit is our body.

Inverse proportion: Two quantities are said to be inversely proportional to each other if the product of their values at any instant is a constant. The increase in one quantity results in a decrease in the other. Example: The more we eat junk, the less is our physical fitness.